Armature: automatic weights not working correctly
Happend to my model as well, exactly as you described.
Weighting couldn't be solved on the model, so I removed entire head and then applied armature with automatic weighting. Weighting was solved as in the bottom image (some weight, though not visible, was all over model). Then I removed eyes, glued head back to the body and weighting was solved normally.
I had bones sticking out of mesh when got this wrong weighting.